The second law of thermodynamics leads to the definition of a new property called entropy, which is a quantitative measure of microscopic disorder for a system. The definition of entropy is based on the Clausius inequality, given by
where the equality holds for internally or totally reversible processes and the inequality for irreversible processes. Any quantity whose cyclic integral is zero is a property, and entropy is defined as
where Sgen is the entropy generated during the process. Entropy change is caused by heat transfer, mass flow, and irreversibilities. Heat transfer to a system increases the entropy, and heat transfer from a system decreases it. The effect of irreversibili- ties is always to increase the entropy.
Entropy is a property, and it can be expressed in terms of more familiar properties through the T ds relations, expressed as
These two relations have many uses in thermodynamics and serve as the starting point in developing entropy-change relations for processes. The successful use of T ds relations de- pends on the availability of property relations. Such relations do not exist for a general pure substance but are available for incompressible substances (solids, liquids) and ideal gases.
The entropy-change and isentropic relations for a process can be summarized as follows:
The work done during a steady-flow process is proportional to the specific volume. Therefore, u should be kept as small as possible during a compression process to minimize the work input and as large as possible during an expansion process to maximize the work output.
The reversible work inputs to a compressor compressing an ideal gas from T1, P1 to P2 in an isentropic (Puk = constant), polytropic (Pun = constant), or isothermal (Pu = constant) manner, are determined by integration for each case with the following results:
The work input to a compressor can be reduced by using multistage compression with intercooling. For maximum savings from the work input, the pressure ratio across each stage of the compressor must be the same.